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Question 1 asks us to write a program that performs delay-coordinate embedding on a time series
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Here is a Matlab implementation of this code
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Notice its very simple: its just two nested for loops that walk over the time series and place the elements in a matrix in the correct order
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If you run this code using a tau of 8 and an m of 7 on the amplitude.dat time series, and we plot the zeroth element versus the second element, we get the following plot
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This object does appear to be a chaotic attractor, so the answer to part a is Chaotic
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Part b states that the dynamics used to generate the trajectory from the last problem was three-dimensional and had a capacity dimension of 2.1
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What requirements does the Takens theorem place on m for a successful embedding of this time series?
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This would be that m needs to be greater than 6
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The Takens theorem says that the m should be chosen greater than 2 times the dimension of the original system, which in this case was three
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So m > 6 is the answer
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Notice that m = 7 would work, because 7 is greater than 6 here
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However, the restrictions on the Takens theorem would state m greater than 6, not m equal to 7